Systems and methods for evaluating the appearance of a gemstone

ABSTRACT

Of the “four C&#39;s,” cut has historically been the most complex to understand and assess. This application presents a three-dimensional mathematical model o study the interaction of light with a fully faceted, colorless, symmetrical round-brilliant-cut diamond. With this model, one can analyze how various appearance factors (brilliance, fire, and scintillation) depend on proportions. The model generates images and a numerical measurement of the optical efficiency of the round brilliant-called DCLR—which approximates overall fire. DCLR values change with variations in cut proportions, in particular crown angle, pavilion angle, table size, star facet length, culet size, and lower girdle facet length. The invention describes many combinations of proportions with equal or higher DCLR than “Ideal” cuts, and these DCLR ratings may be balanced with other factors such as brilliance and scintillation to provide a cut grade for an existing diamond or a cut analysis for prospective cut of diamond rough.

PRIORITY

This application is a divisional application of U.S. patent applicationSer. No. 09/687,759, filed Oct. 20, 2000.

BACKGROUND OF THE INVENTION

The quality and value of faceted gem diamonds are often described interms of the “four C's”: carat weight, color, clarity, and cut. Weightis the most objective, because it is measured directly on a balance.Color and clarity are factors for which grading standards have beenestablished by GIA, among others. Clamor for the standardization of cut,and calls for a simple cut grading system, have been heard sporadicallyover the last 27 years, gaining strength recently (Shor, 1993, 1997;Nestlebaum, 1996, 1997). Unlike color and clarity, for which diamondtrading, consistent teaching, and laboratory practice have created ageneral consensus, there are a number of different systems for gradingcut in round brilliants. As described in greater detail herein, thesesystems are based on relatively simple assumptions about therelationship between the proportions and appearance of the roundbrilliant diamond. Inherent in these systems is the premise that thereis one set (or a narrow range) of preferred proportions for roundbrilliants, and that any deviation from this set of proportionsdiminishes the attractiveness of a diamond. However, no system describedto date has adequately accounted for the rather complex relationshipbetween cut proportions and two of the features within the canonicaldescription of diamond appearance—fire and scintillation.

Diamond manufacturing has undergone considerable change during the pastcentury. For the most part, diamonds have been cut within very closeproportion tolerances, both to save weight while maximizing appearanceand to account for local market preferences (Caspi, 1997). Differencesin proportions can produce noticeable differences in appearance inround-brilliant-cut diamonds. Within this single cutting style, there issubstantial debate—and some strongly held views—about which proportionsyield the best face-up appearance (Federman, 1997). Yet face-upappearance depends as well on many intrinsic physical and opticalproperties of diamond as a material, and on the way these propertiesgovern the paths of light through the faceted gemstone. (Otherproperties particular to each stone, such as polish quality, symmetry,and the presence of inclusions also effect the paths of light throughthe gemstone).

Diamond appearance is described chiefly in terms of brilliance (whitelight returned through the crown), fire (the visible extent of lightdispersion into spectral colors), and scintillation (flashes of lightreflected from the crown). Yet each of these terms cannot be expressedmathematically without making some assumptions and qualifications. Manyaspects of diamond evaluation with respect to brilliance are describedin “Modeling the Appearance of the Round Brilliant Cut Diamond: AnAnalysis of Brilliance.” Gems & Gemology, Vol. 34, No. 3, pp. 158-183(which is hereby incorporated by reference).

Several analyses of the round brilliant cut have been published,starting with Wade (1916). Best known are Tolkowsky's (1919)calculations of the proportions that he believed would optimize theappearance of the round-brilliant-cut diamond. However, Tolkowsky'scalculations involved two-dimensional images as graphical andmathematical models. These were used to solve sets of relatively simpleequations that described what was considered to be the brilliance of apolished round brilliant diamond. (Tolkowsky did include a simpleanalysis of fire, but it was not central to his model).

The issues raised by diamond cut are beneficially resolved byconsidering the complex combination of physical factors that influencethe appearance of a faceted diamond (e.g., the interaction of light withdiamond as a material, the shape of a given polished diamond, thequality of its surface polish, the type of light source, and theillumination and viewing conditions), and incorporating these into ananalysis of that appearance.

Diamond faceting began in about the 1400s and progressed in stagestoward the round brilliant we know today (see Tillander, 1966, 1995). Inhis early mathematical model of the behavior of light in fashioneddiamonds, Tolkowsky (1919) used principles from geometric optics toexplore how light rays behave in a prism that has a high refractiveindex. He then applied these results to a two-dimensional model of around brilliant with a knife-edge girdle, using a single refractiveindex (that is, only one color of light), and plotted the paths of someillustrative light rays.

Tolkowsky assumed that a light ray is either totally internallyreflected or totally refracted out of the diamond, and he calculated thepavilion angle needed to internally reflect a ray of light entering thestone vertically through the table. He followed that ray to the otherside of the pavilion and found that a shallower angle is needed there toachieve a second internal reflection. Since it is impossible to createsubstantially different angles on either side of the pavilion in asymmetrical round brilliant diamond, he next considered a ray thatentered the table at a shallow angle. Ultimately, he chose a pavilionangle that permitted this ray to exit through a bezel facet at a highangle, claiming that such an exit direction would allow the dispersionof that ray to be seen clearly. Tolkowsky also used this limiting caseof the ray that enters the table at a low angle and exits through thebezel to choose a table size that he claimed would allow the most fire.He concluded by proposing angles and proportions for a round brilliantthat he believed best balanced the brilliance and fire of a polisheddiamond, and then he compared them to some cutting proportions that weretypical at that time. However, since Tolkowsky only considered onerefractive index, he could not verify the extent to which any of hisrays would be dispersed. Nor did he calculate the light loss through thepavilion for rays that enter the diamond at high angles.

Over the next 80 years, other researchers familiar with this workproduced their own analyses, with varying results. It is interesting(and somewhat surprising) to realize that despite the numerous possiblecombinations of proportions for a standard round brilliant, in manycases each researcher arrived at a single set of proportions that heconcluded produced an appearance that was superior to all others.Currently, many gem grading laboratories and trade organizations thatissue cut grades use narrow ranges of proportions to classify cuts,including what they consider to be best.

Several cut researchers, but not Tolkowsky, used “Ideal” to describetheir sets of proportions. Today, in addition to systems thatincorporate “Ideal” in their names, many people use this term to referto measurements similar to Tolkowsky's proportions, but with a somewhatlarger table (which, at the same crown angle, yields a smaller crownheight percentage). This is what we mean when we use “Ideal” herein.

Numerous standard light modeling programs have also been long availablefor modeling light refractive objects. E.g., Dadoun, et al., TheGeometry of Beam Tracing, ACM Symposium on Computational Geometry, 1985,p. 55-61; Oliver Devillers, Tools to Study the Efficiency of SpaceSubdivision for Ray Tracing; Proceedings of Pixlm '89 Conference; Pub.Gagalowicz, Paris; Heckbert, Beam Tracing Polygonal Objects, Ed.Computer Graphics, SIGGRAPH '84 Proceedings, Vol. 18, No. 3, p. 119-127;Shinya et al., Principles and Applications of Pencil Tracing, SIGGRAPH'87 Proceedings, Vol. 21, No. 4, p. 45-54; Analysis of Algorithm forFast Ray Tracing Using Uniform Space Subdivision, Journal of VisualComputer, Vol. 4, No. 1, p. 65-83. However, regardless of what standardlight modeling technique is used, the diamond modeling programs to datehave failed to define effective metrics for diamond cut evaluation. Seee.g., (Tognoni, 1990) (Astric et al., 192) (Lawrence, 1998) (Shor 1998).Consequently, there is a need for a computer modeling program thatenables a user to make a cut grade using a meaningful diamond analysismetric. Previously, Dodson (1979) used a three-dimensional model of afully faceted round brilliant diamond to devise metrics for brilliance,fire, and “sparkliness” (scintillation). His mathematical model employeda full sphere of approximately diffuse illumination centered on thediamond's table. His results were presented as graphs of brilliance,fire, and sparkliness for 120 proportion combinations. They show thecomplex interdependence of all three appearance aspects on pavilionangle, crown height, and table size. However, Dodson simplified hismodel calculations by tracing rays from few directions and of fewcolors. He reduced the model output to one-dimensional data by using thereflection-spot technique of Rosch (S. Rosch, 1927, ZeitschriftKristallographie, Vol. 65, pp. 46 -48.), and then spinning that computedpattern and evaluating various aspects of the concentric circles thatresult. Spinning the data in this way greatly reduces the richness ofinformation, adversely affecting the aptness of the metrics based on it.Thus, there is a need for diamond evaluation that comprises fire andscintillation analysis.

SUMMARY OF THE INVENTION

According to one embodiment described herein, a system modelsinteraction of light with a faceted diamond and analyzes the effect ofcut on appearance. To this end, computer graphics simulation techniqueswere used to develop the model presented here, in conjunction withseveral years of research on how to express mathematically theinteraction of light with diamond and also the various appearanceconcepts (i.e., brilliance, fire, and scintillation). The model servesas an exemplary framework for examining cut issues; it includesmathematical representations of both the shape of a faceted diamond andthe physical properties governing the movement of light within thediamond.

One mathematical model described herein uses computer graphics toexamine the interaction of light with a standard (58 facet)round-brilliant-cut diamond with a fully faceted girdle. For any chosenset of proportions, the model can produce images and numerical resultsfor an appearance concept (by way of a mathematical expression). Tocompare the appearance concepts of brilliance, fire, and scintillationin round brilliants of different proportions, we prefer a quantity tomeasure and a relative scale for each concept. A specific mathematicalexpression (with its built-in assumptions and qualifications) that aidsthe measurement and comparison of a concept such as fire is known as ametric. In one embodiment, the metric for fire considers the totalnumber of colored pixels, color distribution of the pixels, lengthdistribution of colored segments (as a function of angular position),density distribution of colored segments, angular distribution ofcolored segments, the distribution of colors over both azimuthal andlongitudinal angle, and/or the vector nature (directionality) of coloredsegments. A more preferred embodiment uses the following metric toevaluate fire: sum (over wavelength) of the sum (over the number of raytraces) of the differential area of each ray trace that exceeds a powerdensity threshold cutoff, multiplied by the exit-angle weighting factor.

This may be calculated as follows:DCLR=Σ _(wavelengths)Σ_(rays)(dArea*σ*Weighting Factor).

In this preferred embodiment, if the power density of a trace is greaterthan the threshold cutoff, σ=1; otherwise σ=0 and the ray (or otherincident light element) is not summed. In a most preferred embodiment,comprising a point light source, the metric considers the total numberof colored pixels (sum of rays), the length distribution of coloredsegments (because with a point source, length approximates differentialarea), angular distribution of colored segments (the weighting factor)and a threshold cutoff (σ=0 or 1) for ray (or other incident lightelement) power density. Although other factors (e.g., bodycolor orinclusions) may also influence how much fire a particular diamondprovides, dispersed-color light return (DCLR) is an important componentof a diamond fire metric.

The systems and methods described herein may further be used tospecifically evaluate how fire and scintillation are affected by cutproportions, including symmetry, lighting conditions, and other factors.In addition to the cut proportions expressly including in the tables,other proportions, such as crown height and pavilion depth may bederived from the tables, and used as the basis for optical evaluationand cut grade using the methods and systems disclosed herein. Otherembodiments and applications include an apparatus and system to grade afaceted diamonds, new methods of providing target proportions forcutting diamonds, new types of diamonds cuts and new methods for cuttingdiamonds.

Within the mathematical model, all of the factors considered importantto diamond appearance—the diamond itself, its proportions and facetarrangement, and the lighting and observation conditions—can becarefully controlled, and fixed for a given set of analyses. However,such control is nearly impossible to achieve with actual diamonds. Thepreferred model described herein also enables a user to examinethousands of sets of diamond proportions that would not be economicallyfeasible to create from diamond rough. Thus, use of the model allows theuser to determine how cut proportions affect diamond appearance in amore comprehensive way than would be possible through observation ofactual diamonds. In one preferred embodiment, the system, method andcomputer programs use to model the optical response of a gemstone useHammersley numbers to choose the direction and color for each element oflight refracted into a model gemstone (which defines the gemstonefacets) to be eventually reflected by the model gemstone's virtualfacets, and eventually exited from the model gemstone to be measured bya model light detector. The gemstone is then ultimately graded for itsoptical properties based on the measurement of said exited lightelements from the gemstone model.

In another preferred embodiment, the system determines the grade of acut using certain assumptions—best brilliance, best fire, best balanceof the two, best scintillation, best weight retention, bestcombination—that can be achieved from a particular piece of rough. Inaddition, an instrument may also measure optical performance in realdiamonds based on the models described. The models of light diamondinteraction disclosed herein can also be used to compare and contrastdifferent metrics and different lighting and observation conditions, aswell as evaluate the dependence of those metrics on proportions,symmetry, or any other property of diamond included in the model.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a drawing and table that outlines the assumptions on which apreferred model is based. Diamond model reference proportions in thispatent application, unless otherwise specified, are table 56%, crownangle 34°, pavilion angle 40.5°, girdle facet 64, girdle thickness 3.0%,star facet length 50%, lower girdle length 75%, culet size 0.5%.

FIGS. 2A to 2C are a plot of DCLR versus crown angle over threethresholds for a modeled round brilliant diamond along with the table ofcorresponding data.

FIGS. 3A to 3C are a plot of DCLR versus pavilion angle over threethresholds for a modeled round brilliant diamond along with the table ofcorresponding data.

FIGS. 4A to 4C are a plot and table of DCLR with reference to crownangle and table size for a low power density threshold cutoff modelingsystem.

FIGS. 5A to 5C are a plot and table of DCLR with reference to crownangle and table size for a medium power density threshold cutoffmodeling system.

FIGS. 6A to 6C are a plot and table of DCLR with reference to crownangle and table size for a high power density threshold cutoff modelingsystem.

FIGS. 7A to 7B are a table of DCLR rating for various diamondproportions, varying by star facet length, for 3 values of crown angle.

FIG. 8 is a table of DCLR ratings for various diamond proportions,varying by star facet length, for a medium power density thresholdcutoff modeling system.

FIG. 9 is a table of DCLR ratings for various diamond proportions,varying by star facet length, for a low power density threshold cutoffmodeling system.

FIGS. 10A to 10F are a table of DCLR ratings for various diamondproportions, varied by pavilion angle and table size, for a high powerdensity threshold cutoff modeling system.

FIGS. 11A to 11F are a table of DCLR ratings for various diamondproportions, varied by pavilion angle and table size, for a medium powerdensity threshold cutoff modeling system.

FIGS. 12A to 12F are a table of DCLR ratings for various diamondproportions, varied by pavilion angle and table size, for a low powerdensity threshold cutoff modeling system.

FIG. 13 is a diagram of one fourth of the view from infinity of thetotally dispersed light for a diamond of 33.5° crown angle, 4.00pavilion angle, and table 0.55 with 64 girdle facets, a 3% girdlethickness, a 50% star facet length, 75% lower-girdle length and a 0.5%culet size.

FIG. 14 is a diagram of one fourth of the view from infinity of thetotally dispersed light for a diamond of 31.5° crown angle, 38.7°pavilion angle, and table 0.52 with 64 girdle facets, a 3% girdlethickness, a 50% star facet length, 75% lower-girdle length and a 0.5%culet size.

FIG. 15 is a diagram of one fourth of the view from infinity of thetotally dispersed light for a diamond of 31.5° crown angle, 40.7°pavilion angle, and table 0.52 with 64 girdle facets, a 3% girdlethickness, a 50% star facet length, 75% lower-girdle length and a 0.5%culet size.

FIG. 16 is a diagram of one fourth of the view from infinity of thetotally dispersed light for a diamond of 31.5° crown angle, 42.7°pavilion angle, and table 0.52 with 64 girdle facets, a 3% girdlethickness, a 50% star facet length, 75% lower-girdle length and a 0.5%culet size.

FIG. 17 is a diagram of one fourth of the view from infinity of thetotally dispersed light for a diamond 33.5° crown angle, 40.7° pavilionangle, and table 0.60 with 64 girdle facets, a 3% girdle thickness, a50% star facet length, 75% lower-girdle length and a 0.5% culet size.

FIG. 18 is a diagram of one fourth of the view from infinity of thetotally dispersed light for a diamond 35.3° crown angle, 40.0° pavilionangle, and table 0.56 with 64 girdle facets, a 3% girdle thickness, a50% star facet length, 75% lower-girdle length and a 0.5% culet size.

FIG. 19 is a diagram of one fourth of the view from infinity of thetotally dispersed light for a diamond 28.5° crown angle, 40.7° pavilionangle, and table 0.53 with 64 girdle facets, a 3% girdle thickness, a50% star facet length, 75% lower-girdle length and a 0.5% culet size.

FIG. 20 is a diagram of one fourth of the view from infinity of thetotally dispersed light for a diamond 28.5°, crown angle, 40.7° pavilionangle, and table 0.63 with 64 girdle facets, a 3% girdle thickness, a50% star facet length, 75% lower-girdle length and a 0.5% culet size.

FIG. 21 is a diagram of one fourth of the view from infinity of thetotally dispersed light for a diamond 34.5°, crown angle, 40.7° pavilionangle, and table 0.57 with 64 girdle facets, a 3% girdle thickness, a50% star facet length, 75% lower-girdle length and a 0.5% culet size.

FIG. 22 is a diagram of one fourth of the view from infinity of thetotally dispersed light for a diamond 32.7°, crown angle, 41.50 pavilionangle, and table 0.60 with 64 girdle facets, a 3% girdle thickness, a50% star facet length, 75% lower-girdle length and a 0.5% culet size.

FIG. 23 is a table of DCLR rating for certain diamond proportions,varying by table size.

FIG. 24 is a table of DCLR rating for certain diamond proportions,varying by lower girdle size.

FIG. 25 is a plot of DCLR versus culet size corresponding to FIGS. 26Aand 26B.

FIGS. 26A and 26B are a table of DCLR rating for certain diamondproportions, varying by culet size.

DESCRIPTION OF THE INVENTION

Assumptions and Methods. The mathematical model presented here creates afresh structure for examining nearly all aspects of the influence thatcut has on a diamond's appearance. FIG. 1 provides the assumptions onwhich a preferred model may be based: a detailed list of the physicalproperties included in the model, a mathematical description of theproportions of the round brilliant, and a description of the lightingcondition used in this study. The details of the lighting conditionsaffect the specific numerical values we present here. The model traceslight from the modeled light source through a mathematicalrepresentation of a round brilliant of any chosen proportions (referredto hereafter as the “virtual” diamond) to produce two kinds of results:(1) digital images of the virtual diamond, and (2) a numericalevaluation of an appearance concept (in this case, fire).

The metrics disclosed herein may be run on any computer, such as aPentium-based PC using standard light refraction modeling techniques andlight elements, including those used in CAD Programs, as are known inthe art.

The preferred metric for fire, Dispersed Colored Light Return (DCLR), isan original product the development of which required considerablecreative thought. DCLR describes the maximum extent to which a given setof proportions can disperse light toward an observer; the value isdefined using a point light source at infinite distance and ahemispherical observer also located at infinity. (In general, observeddispersion depends strongly on the light source and observationgeometry: as the distance between the observer and diamond increases,the observer sees less white light and more dispersed colors).

Another metric, describing scintillation, may consider both the staticview (amount and degree of contrast) and the dynamic view (how thecontrast pattern changes with movement), and may factor in parts ofbrilliance (how the spatial resolution of the contrast interacts withhuman vision to affect how “bright” an object looks, and the effects ofglare), and describe what most diamond cutters call “life,” and Dodson(1979) calls “sparkliness.” The relevant scintillation factors for thestatic view include the number of edges seen across the face of theround brilliant, the distribution of distances between those edges, theshapes made by them, the contrast in output power across those edges(e.g. black against white or medium gray against pale gray), and thevisual impact of colored rays on the appearance of the black and whitepattern. All these aspects are present in the “view-from infinity” (VFI)diagrams of the model output; See FIGS. 13-22, however, they are alsodiscernable in a head-on photo or direct observation of a diamond. Therelationship between the positions of exit rays at infinity and theshapes they form on an image plane above the stone (parallel to thetable) at some distance, enables a user of the model to calculate ascintillation metric from the raw data at any chosen distance. Thefactors listed above change in numerical value with differences invertical distance. Thus, the metric may be based on a vertical distanceor distances suitable to approximate the experience of a standardobserver.

The metrics for fire and scintillation may also incorporate dynamicaspects. Dynamic aspects into the preferred fire metric, DCLR, areobtained by placing the observer at infinity and weighting thecontributions of rays by their exit angle with a cosine-squaredfunction. Another way to explore dynamic shifts is to move the lightsource—such that the incoming rays are perpendicular to a bezel or starfacet rather than the table, and compare the output (both the diagramand DCLR value) to that obtained with the light source directly over thetable. The dynamic aspects of scintillation likewise involve changes inthe black-and-white pattern with motion of the stone, light source, orobserver.

The details of human vision may also be incorporated in each of thesemetrics. Thus, DCLR preferably incorporates a threshold for theamplitude range of human vision with “ordinary” background illumination.(Humans see considerably more than the 256 levels of gray used by acomputer monitor). The scintillation metric incorporates human visionaspects related to contrast intensity and spatial resolution ofcontrasting light levels and colors and considers how colored rays lookagainst different patterns. These aspects of human vision also come intoplay in the design of a human observation exercise, wherein a number ofpeople will observe a fixed set of diamonds under one or more fixedviewing conditions, and compare their brilliance, brightness, fire, andscintillation, as a check on the predictions from modeling.

Although the human visual system can detect as few as 7 photons when itis fully adapted to the dark, far more light is required to stimulate aresponse in an ordinarily bright room. The specific range of the humanvisual system in ordinary light has not been definitively measured, butprofessional estimates suggest detection of up to 10,000 gray levels. (Acomputer monitor uses 256 levels, and high-quality photographic film hasjust under 1000). Thus it is uncertain how much of fire to take intoconsideration to match the capacity of human vision: Accordingly, oneembodiment of the metric comprises a threshold power density cutoff toapproximate human vision. Furthermore, the power density threshold maybe weighted to account for differentiation in human eye sensitivity todifferent parts of visual spectrum (e.g., use a higher threshold cutofffor green light because humans have lower sensitivity for green ascompared to blue light). This principle also applies with force to thescintillation metric. As disclosed herein, DCLR values may be calculatedusing ranges of 2, 3, and 4 orders of magnitude (i.e. including raysdown to 100 (fire 2), 1000 (fire 3), and 10,000 (fire 4) times weakerthan the brightest ones). In the preferred embodiment, DCLR is adirectly computed value, and traces all light from the source so thereis no convergence and no error. The results are shown as DCLR valuesgraphed against various proportion parameters. See FIGS. 2A to 2Cthrough 6A to 6C. Fire 2 means that a threshold eliminates refractedlight elements at less than 1% of the brightest light elements. Fire 3uses a cut off of 0.1% off and Fire 4 uses a 0.01% cut off. The obviousresult from this initial data is that DCLR (and thus fire) does not havea monotonic dependence on only the crown proportions, as Tolkowsky's1919 work claimed, but shows a multi-valued dependence on severalproportions, including the pavilion angle. In other words, DCLR, likeWLR, can be maximized in a number of ways.

Different lighting geometries emphasize different aspects of a diamond'sappearance. Thus, although the lighting and observing conditions must bespecified for a given metric, these conditions can be varied and used incalculation of similar metrics.

Likewise, in a preferred embodiment, the model assumes a fully facetedgirdle, perfect symmetry, perfect polish, no color, no fluorescence, noinclusions, and no strain. Actual diamonds may have bruted girdles,asymmetries (e.g. culet off center, or table not parallel to girdle),scratches and polishing lines, color, blue or yellow fluorescence ofvarying strengths, a variety of inclusions, and a strain in a variety ofdistributions. Each of these properties affects the movement of lightand the actual expression of the appearance aspects. Many of theseaspects may be incorporated into the model. In another embodiment, theinvention contemplates the use of a device (or devices, one for eachmetric) that measures the various appearance metrics for actualdiamonds, including each one's particular oddities.

Although the DCLR may be calculated for the idealized set of averageproportions, they may also be calculated for that of a particular stone.Thus, in another embodiment, a low end grade may be used for the diamondindustry and jewelers; the metrics disclosed herein readily identifysets of proportions with poor optical performance. See FIGS. 2A to 2Cthrough 6A to 6C.

Defining Metrics: FIRE.

One advantage of using a computer model is the capability it gives us toexamine thousands of proportion variations. To make sense of so muchdata, however, we needed to define a metric for fire, and use it tocompare the performance of the different proportion combinations. Avariety of mathematical expressions can be created to describe suchlight. Each expression requires explicit or implicit assumptions aboutwhat constitutes fire and about light sources, viewing geometry,response of the human eye, and response of the human brain. Themathematical definition of fire may represent one viewing geometry—thatis, a “snapshot”—or, more preferably, represent an average over manyviewing situations.

Dispersed-Colored Light Return. A preferred metric described herein iscalled Dispersed Colored Light Return (DCLR); it is specific to each setof modeled diamond proportions with the chosen illumination. Afterexamining a variety of possible metrics for fire, DCLR represents thebest way to evaluate fire using a viewing model that looks at the stonefrom an infinite distance to achieve maximum dispersion.

According to this preferred embodiment, the metric for fire, DCLR, usesan approach that is completely different than the approach Dodson (1979)used. Starting with a point light source at infinity and a hemisphericalobserver, also at infinity, the preferred metric takes into account thesize, brightness, exit angle, number and color of all incident lightelements that exit the crown using the following equation:DCLR=Σ _(wavelengths)Σ_(light elements)(dArea*Weighting Factor).

In a more preferred embodiment, the method uses the same weightingfactor, the square of the cosine of the exit angle, as in the WeightedLight Return Model discussed in Gems and Gemology Vol. 34, No. 3. pp.158-183, Fall 1998 (e.g. rays that exit the modeled diamond vertically(90%) have a weighting factor of 1, and rays that exit at 65° have aweighting factor of 0.82). This weighting numerically mimics the commonindustry practice of rocking a stone back and forth and from side toside while observing it, through an angular sweep of about 35-40% fromthe vertical. The light elements may be pencils, bundles, rays or anyother light unit element known in the light modeling art.

The light elements to be included in DCLR may be also required to meet apower density threshold cutoff. Thus, in a most preferred embodiment,the DCLR is a sum (over wavelength) of the sum (over the number of lightelement traces) of the differential area of each light element tracethat surpasses a threshold power density cutoff (most preferably 1% ofthe brightest element) times an exit-angle weighting factor.

The most preferred embodiment may beneficially trace pencils of lightforward through the gemstone model and then trace rays backwards throughthe model to measure the optical properties of a gemstone. Each of thegemstone illumination models used herein may also include the use ofHammersley numbers to determine the direction and color for each lightelement directed at the gemstone model.

Dodson (1979) evaluated his metrics for 3 crown heights (10, 15, and20%), 4 table sizes (40, 50, 60, and 70%), and 10 pavilion anglesbetween 38 and 55%, a total of 120 proportion combinations, and showedthat his three metrics yielded wide variations across these proportions.In contrast, the present description includes a calculated DCLR for 2148combinations of 6 proportions: crown angle, pavilion angle, table size,star facet length, lower girdle length, and culet size. (This rangeincludes both common commercial proportions and values of crown anglesand star facet lengths that are very rarely cut). See FIGS. 7A and 7Bthrough 12A to 12F. These metrics are computed functions of the 8independent shape variables, and each data set forms a surface over the6 shape variables we have varied to date. We have explored thetopography of the DCLR surface with standard graphical and numericaltechniques, to find all those combinations that yield high DCLR, and toreveal relationships between proportions and brightness.

Moreover, using previously published WLR data, a user can also comparethe DCLR data set with the previously described Weighted Light Returnset (see Gem & Gemology Vol. 34, No. 3, pp. 158-183) or other brilliancedata to find proportions that yield an attractive balance of brillianceand fire.

Results

In the preferred model, a point light source at infinite distance shineson the table of a virtual diamond of chosen proportions; because thelight source is so far away all the entering rays are parallel. Theserays refract and reflect, and all those that refract out of the crownfall on the observer, a hemisphere at infinite distance. Because theobserver is so far away, all the light that falls on it is fullydispersed; thus, there is no “white” output. DCLR results are shown inFIGS. 2-12. The VFI diagrams are direct output resulting from the model,with the background color reversed from black to white for greater easein viewing and printing. See FIGS. 13-22. A VFI diagram is one fourth ofthe observer hemisphere, unrolled onto the page or screen; the point isthe overhead center of the hemisphere (light exiting perpendicular tothe table, and the rounded border is the edge of the hemisphere (lightexiting parallel to the girdle).

All static aspects of fire and scintillation are contained within thisoutput. However, of the qualities we considered relevant to fire; only 3of those 7 ended up in the most preferred metric (total number, lengthdistribution [changed to differential area], and angular distribution)and we added a new concept, that of the threshold for power density.That concept comes from making the VFI diagrams because the number ofcolored segments changed so noticeably as a function of power density.

Images and DCLR. The calculations made with our model also may be usedto produce realistic digital images of virtual diamonds. Thus,computer-generated images can reproduce the patterns of light and darkseen in actual round brilliant diamonds under lighting conditionssimilar to those used with the model. The model can generate a varietyof digital images, from different perspectives and with differentlighting conditions. However, the details of how fire changes withproportions can be better studied by comparing a metric, such as DCLRvalues, than by visually examining thousands of images, whether VFIdiagrams or virtual diamonds themselves.

Results for Key Individual Parameters. Our investigation of thedependence of DCLR on crown angle, pavilion angle, star facet length,and table size, began with an examination of how DCLR varies with eachof these three parameters while the remaining seven parameters are heldconstant. Except where otherwise noted, we fixed these parameters at thereference proportions (see FIG. 1). See FIGS. 7A and 7B through 12A to12F.

Crown Angle. In general, DCLR increases as crown angle increases; but,as FIGS. 2A to 2C show, there are two local maxima in DCLR across therange of angles, at about 25° and 34-35°, and a rise in values at crownangles greater than 41°. However, moderately high crown angles of 36-40°yield a lower DCLR value than either of the local maxima. The sametopography is seen at each of the three thresholds, although thenumerical range of each data set (the difference between the maximum andminimum values) decreases as the threshold is raised.

Pavilion Angle. This is often cited by diamond manufacturers as theparameter that matters most in terms of brilliance (e.g., G. Kaplan,pers. comm., 1998), but we surprisingly found the greatest variation inDCLR for changes in pavilion angle. FIGS. 3A to 3C show an overalldecrease in DCLR (calculated with the lowest threshold) with increasingpavilion angle, with a true maximum at 38.75°, and local maxima at40-41° and 42.25°. Unlike crown angle, pavilion angles are typicallymanufactured in a fairly narrow range; the peak from 40-41° covers abroad range for this parameter. Similar topography is seen for theintermediate threshold, but the peak at low pavilion angle is absentfrom DCLR calculated at the highest threshold.

Star Facet Length. We calculated the variation of DCLR (with the lowestthreshold) with changes in the length of the star facet for three valuesof the crown angle: 34°, 36°, and 25°. The range in DCLR values isrelatively small, but as seen in FIGS. 7A and 7B, 8, and 9 there is aprimary maximum in each array. At the reference crown angle of 34°, astar facet length of 0.56 yields the highest DCLR. This maximum shiftsto about 0.58 for a crown angle of 36°, and increases substantially to astar facet length of 0.65-0.65 for a crown angle of 25°. Longer starfacet length means that the star facet is inclined at a steeper anglerelative to the table (and girdle, in a symmetrical round brilliant),and thus these results imply that the star facets act similarly to thebezel facets with regard to the production of fire. Also, as with crownangle, similar topography is seen in the arrays calculated with higherthresholds but with significantly reduced range of DCLR values.

Two of the high-threshold arrays (34° and 36° crown angle) and themedium-threshold data show secondary maxima at star facet lengths of0.3, 0.32 and 0.36 respectively. Neither such short stars, nor thelonger stars indicated by the primary maxima, are commonly used in theproduction of round brilliant diamonds.

Table Size. DCLR shows a bi-modal response to variations in table size,as shown in FIGS. 10A to 10F, 11A to 11F, and 12A to 12F. For the lowand medium thresholds, DCLR is approximately constant for tables lessthan 0.55, rapidly decreases for tables of 0.56 and 0.57, and thenremains approximately constant for tables of 0.58 and greater. For thehighest threshold, DCLR is approximately constant across the entirerange of table sizes. See, e.g., FIG. 23.

Lower Girdle. The variation of DCLR with lower girdle facet length ismoderate, similar in magnitude to the variation found with crown angle.For all three thresholds, longer lower girdle facets are favored, withbroad maxima at 0.80-0.85. Lower girdle facets form an angle with thegirdle plane that is less than the pavilion angle; the longer thesefacets are the closer their angle becomes to the pavilion angle. SeeFIG. 24.

Culet Size. Unlike WLR, which showed little dependence on culet size,DCLR decreases significantly with increasing culet size. This decreaseis smooth and monotonic, and for the lowest threshold the DCLR valuedecreases by 25%. See FIGS. 25, 26A and 26B.

Thus, as shown in the tables and figures disclosed herein, a cut gradethat considers fire can be made by reference to enter star facet length,lower girdle length, and culet size. For example, as shown in FIGS. 2-6,the cut grade may be based on a fire peak within 40-41° pavilion angle,but also recognize fire peaks substantially at 38.75° and 42.5°.

Combined Effects. Some of the interactions between crown angle, pavilionangle, and table size—and their combined effects on DCLR values—can beseen when these proportion parameters are examined two at a time. Oneway to visualize these effects is to draw them to look like atopographic map (which shows the differences in elevation of an area ofland). We can draw subsets of the data as cross-sections (slices)through the data set with one parameter held constant, and the WLRvalues can then be expressed as contours. These cross-sections can beread in the same manner as topographic maps; but instead of mountains,these “peaks” show proportion combinations that produce the highestcalculated DCLR values.

FIGS. 4A to 4C show such a contour map for DCLR (calculated with thelowest threshold) with variation in both crown angle and table size. Two“ridges” of rapidly varying DCLR values are evident at crown angles of25-26° and crown angles greater than or equal to 34°. This latter ridgeis broad and shows convoluted topography. These ridges become gullieswith decreasing table size; that is, at these crown angles, table sizesof 0.58 and less yield high DCLR values, but larger table sizes yieldlower DCLR values than are found at other crown angles. In particular,there is a local maximum in DCLR for tables of 0.65-0.63 and a crownangle of 29°.

Somewhat similar topography is observed in FIGS. 5 and 6, contour mapsof DCLR over crown angle and table size for the medium and highthresholds, respectively. At the medium threshold, crown angles of37-38° yield significantly lower DCLR at all table sizes greater than0.57, while crown angles of 32-33° yield moderate DCLR across the wholerange of table sizes. There is a large ridge across shallow crown anglesand all table sizes in the plot for the highest threshold, although forthis data the numerical range of the values is quite small.

FIGS. 10A to 10F, 11A to 11F and 12A to 12F give the data for variationin DCLR as pavilion angle and table size each vary, for the threethresholds. The topography becomes much more complex as the threshold islowered, and the range of values increases considerably. For the lowestthreshold, there is a small ridge at a pavilion angle of 38.25 and tablesizes of 0.56 and lower, and for all three thresholds there is a longridge at a pavilion angle of 39.25 across the whole range of tablesizes. This ridge appears more broad at the highest threshold, coveringpavilion angles from 39-41′.

Importantly, the FIGS. 4A to 4C through 6A to 6C and 10A to 10F through12A to 12F demonstrate that preferred “fire” proportions based on thedisclosed proportion parameters can serve as guides or even ranges in acut grade determination.

Using DCLR Data to Evaluate Fire. The DCLR surfaces that we havecalculated as a function of crown angle, pavilion angle, and table sizeare irregular, with a number of maxima, rather than a single maximum.These multiple “peaks” are a principal result of this extensivethree-dimensional analysis. Their existence supports a position taken bymany in the trade in terms of dispersed light return, or fire there aremany combinations of parameters that yield equally “attractive” roundbrilliant diamonds. Neither the internal dispersion of light nor theinteraction between the proportion parameters is taken into account byexisting cut-grading systems, which are based on Tolkowsky's analysis ata single refractive index, and examine each parameter separately.

It is especially important to note that some proportion combinationsthat yield high DCLR values are separated from one another and notcontiguous, as shown in the cross-sections of the DCLR surfaces. Thus,for some given values of two proportions, changes in the thirdproportion in a single direction may first worsen DCLR and then improveit again. This variation in DCLR with different proportion combinationsmakes the characterization of the “best” diamonds, in terms of fire, agreat challenge. Even for one simple shape—the round brilliant cut—andvariation of only two proportion parameters at a time, the surfaces ofconstant DCLR are highly complex.

The specific proportion combinations that produce high DCLR values havea variety of implications for diamond manufacturing. Because manycombinations of proportions yield similarly high DCLR values, diamondscan be cut to many choices of proportions with the same fire, whichsuggests a better utilization of rough.

Evaluation of “Superior” Proportions Suggested by Earlier Researchers. Agem diamond should display an optimal combination of brilliance, fire,and pleasing scintillation. Many previous researchers have suggestedproportions that they claim achieve this aim, but none but Dodson haveproposed a measure or test to compare the fire or scintillation of twosets of proportions. A list of “superior” proportions and theircalculated WLR value was presented in Hemphill et al. (1998), and wehave calculated DCLR for some of these proportions as well. The highestvalue we found was for Suzuki's Dispersion Design (1970), with a DCLR(at the lowest threshold, as are all the values presented in thisdiscussion) of 6.94; however this set of proportions had yielded a verylow WLR value of 0.205. Eppler's Ideal Type II proportions yielded arelatively high DC-LR value of 5.04, and a moderately high WLR value of0.281. Dodson's suggestion for most fiery was bright (WLR=0.287) butyielded a low DCLR of 4.32. Dodson's proportions for the mostsparkliness yielded a higher DCLR of 5.18, but with a low WLR value of0.247. His suggestion for brightest had yielded an average WLR of 0.277,and a moderately low DCLR of 4.51.

Work by Shannon and Wilson, as described in the trade press (Shor,1998), presented four sets of proportions that they claimed gave“outstanding performance” in terms of their appearance. Previously wecalculated typical to moderately high WLR values for these proportions,and now we find moderate to moderately high DCLR values of 4.63-5.24. Incomparison, Rosch's suggestion for “Ideal” proportions had yielded a lowWLR value of 0.251, but produce high DCLR of 5.94. Tolkowsky's suggestedproportions, including the knife-edge girdle and a 53% table, yield aDCLR value of 5.58, but this value is reduced significantly as the tablesize or girdle thickness increases.

Implications for Existing Cut-Grading Systems. Our results disagree withthe concepts on which the proportion grading systems currently in use byvarious laboratories appear to be based. In particular, they do notsupport the idea that all deviations from a narrow range of crown anglesand table sizes should be given a lower grade. Nor do they support thepremises that crown proportions matter most for fire.

Arguments that have been made for downgrading diamonds with lower crownangles or larger tables on the basis that they do not yield enough fireare in part refuted by the results of our modeling. Our results showmore agreement with those of Dodson (1979): that fire depends oncombinations of proportions, rather than on any single parameter.However, our results are at a finer scale than those of Dodson, and showdistinct trends for certain ranges of proportion combinations.

REFERENCES

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W. (1936) New diamond cuts break more easily. Gems &    Gemology, Vol. 2, No. 4, p. 68.-   Watermeyer B. (1991) Diamond Cutting, 4th ed. Preskor Doomfontein,    Johannesburg, 406 pp. Wright, W. D. (1969) The Measurement of    Colour, 4th ed., Van Nostrand, New York.    Box A:    Detailed Description of One Diamond Model Embodiment

In one embodiment, the diamond model describes a faceted diamond as aconvex polyhedron, a three-dimensional object with a surface that isbounded by flat planes and straight edges, with no indentations orclefts. The model requires that all surfaces be faceted, including thegirdle, and currently excludes consideration of indented naturals orcavities. To date, we have focused our calculations on the roundbrilliant cut because of its dominant position in the market, but thismodel can be used for nearly any fully faceted shape. Our modeled roundbrilliant has mathematically perfect symmetry; all facets are perfectlyshaped, pointed, and aligned. Also, all facet junctions are modeled withthe same sharpness and depth.

Because our modeled round brilliant has perfect eight-fold symmetry,only eight numbers (proportion parameters) are required to specify theconvex polyhedron that describes its shape (FIG. A-1). (Modeling othershapes or including asymmetries requires additional parameters). Wedefined these eight parameters as: Crown angle Angle (in degrees)between the bezel facets and the girdle plane Pavilion Angle (indegrees) between the pavilion mains and the angle girdle plane Tablesize Table width (as percent of girdle diameter) Culet size Culet width(as percent of girdle diameter) Star facet The ratio of the length ofthe star facets to the distance length between the table edge and girdleedge, as projected into the table plane Lower-girdle The ratio of thelength of the lower-girdle facets to the length distance between thecenter of the culet and girdle edge, as projected into the table planeGirdle Measured between bezel and pavilion main facets (the thickthickness part of the girdle) and expressed as a percentage of girdlediameter. This differs from the typical use of the term girdle thickness(see, e.g., GIA Diamond Dictionary, 1993) Girdle facets Total number ofgirdle facets

Other proportions, such as the crown height, pavilion depth, and totaldepth (expressed as percentages of the girdle diameter) can be directlycalculated from these eight parameters, using these formulas:Crown height=1 2(100−table size)×tan(crown angle)Pavilion depth=1 2(100−culet size)×tan(pavilion angle)Total depth=(Crown height+pavilion depth+girdle thickness)

For the results in this application, the diamond simulated in ourcalculations (called a “virtual” diamond) has no inclusions, isperfectly polished, and is completely colorless. It has a polishedgirdle, not a bruted one, so that the girdle facets refract light raysin the same way that other facets do. The virtual diamond isnon-dimensionalized, i.e. it has relative proportions but no absolutesize—that is, no specific carat weight. The principles governing the waylight moves through a colorless diamond do not vary with size, but someaspects of viewing a diamond do depend on its absolute size. A specificdiameter can be applied to the virtual diamond for such purposes, or forothers such as the application of a color or fluorescence spectrum.

We then chose modelled light sources to illuminate our virtual diamond.Results for brilliance (Hemphill et at., 1998) used a diffuse hemisphereof even, white light (D65 daylight illumination) shining on the crown.That illumination condition averages over the many different ambientlight conditions in which diamonds are seen and worn, from the basictrading view of a diamond face-up in a tray next to large north-facingwindows, to the common consumer experience of seeing a diamond wornoutdoors or in a well-lit room. Such diffuse illumination emphasizes thereturn of white light, but it is a poor lighting condition for examiningother fire and scintillation. These aspects are maximized by directedlight, such as direct sunlight or the small halogen track lights commonin many jewelry stores. Directed light is readily modeled as one or morepoint light sources at infinity or as a collimated finite-size spot atsome other distance. For calculation of DCLR we used a D65 point lightsource at infinite distance, centered over the table. This illuminationcondition samples the maximum extent to which the round brilliant candisperse light. This same modelled lighting can be used to examine someaspects of scintillation, although other aspects, particularly dynamicones, will require more than one lighting position.

Next we examined mathematically how millions of rays of light from thesource interact with the transparent, three-dimensional, colorless,fully faceted round brilliant specified by our choice of proportionparameters. Diamond is a dispersive material; the refractive index isdifferent for different wavelengths of light. Since the angle ofrefraction depends on the refractive index, white light entering thevirtual diamond is spread (dispersed) into rays of different colors, andeach of these variously colored rays takes a slightly different paththrough the stone. We used Sellmeier's formula (see Nassau, 1983 [p.211]; or, for a more thorough explanation, see Papadopoulos andAnastassakis, 1991) to incorporate this dispersion into the model. Withthis formula, we obtained the correct refractive index for each of thedifferent colored rays (taken at 1 nm intervals from 360 to 830 nm), sothat each ray could be traced (followed) along its correct path as itmoved through the stone. Very few rays follow simple paths with only afew internal reflections; most follow complex three-dimensional paths(FIG. A-2).

Each time a ray strikes a facet, some combination of reflection andrefraction takes place, depending on the angle between the ray and thefacet, the refractive index at the wavelength of the ray, and thepolarization state of the ray. Although the rays from our point lightsource are initially unpolarized, a light ray becomes partly polarizedeach time it bounces off a facet. The degree and direction ofpolarization affect how much of the ray is internally reflected, ratherthan refracted out the next time it intersects a facet. (For example,about 18% of a light ray approaching a diamond facet from the inside atan angle of 5° from the perpendicular is reflected, regardless of thepolarization. But at an incidence of 70°, rays with polarizationparallel to the plane of incidence are completely lost from the stone,while 55% of a ray polarized perpendicular to the plane of incidence isreflected back into the stone). The model traces each ray until 99.95%of its incident energy has exited the diamond. The end result of thisray tracing can be an image of the virtual diamond (seen from a shortdistance or from infinity) or the value of a metric for that stone.

Although modifications and changes may be suggested by those skilled inthe art, it is the intention of the inventors to embody within thepatent warranted hereon all changes and modifications as reasonable andproperly come within the scope of their contribution to the art.

1. A method of providing target proportions for cutting a diamondcomprising: determining the size of an uncut gemstone; comparing a listof possible cut proportions of the gemstone with a list of proportiongrades that depend, at least in part, on a calculation of dispersedcolor light return; providing a target proportion for the gemstone basedon said list of proportion grades.
 2. A method of cutting a diamondcomprising: determining the size of an uncut diamond; determining apossible cut proportion for the diamond; comparing said possible cutproportion with a list of proportion grades that depend, at least inpart, on a calculation of dispersed color light return; cutting theuncut diamond. 3-110. (canceled)